Question

the percentage of adolescents ages 13 to 18 owning a cell phone in 2012 compared to 2022 for 12 states was gathered.

state	percentage of adolescents with cell phones in 2012	percentage of adolescents with cell phones in 2022
2	15.3	27.1
3	16.8	27.4
4	19	28.9
5	21.1	31.7
6	21.3	41.1
7	21.4	40
8	21.5	42
9	22.1	50.9
10	24.6	43.7
11	28.7	52.6
12	30.8	72.3

what is the predicted percentage of adolescents having a cell phone in 2022 for state 13 if the percentage in 2012 was 23.8?

Explanation:

Step1: Calculate regression slope

First, compute the mean of 2012 values ($\bar{x}$) and 2022 values ($\bar{y}$):
$\bar{x} = \frac{11.9+15.3+16.8+19+21.1+21.3+21.4+21.5+22.1+24.6+28.7+30.8}{12} = \frac{254.5}{12} \approx 21.21$
$\bar{y} = \frac{25.9+27.1+27.4+28.9+31.7+41.1+40+42+50.9+43.7+52.6+72.3}{12} = \frac{503.6}{12} \approx 41.97$

Next, calculate the slope $m$ using:
$m = \frac{\sum_{i=1}^{12}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{12}(x_i-\bar{x})^2}$

Compute numerator terms:
$\sum(x_i-\bar{x})(y_i-\bar{y}) = (11.9-21.21)(25.9-41.97)+(15.3-21.21)(27.1-41.97)+...+(30.8-21.21)(72.3-41.97) \approx 692.39$

Compute denominator terms:
$\sum(x_i-\bar{x})^2 = (11.9-21.21)^2+(15.3-21.21)^2+...+(30.8-21.21)^2 \approx 322.68$

$m = \frac{692.39}{322.68} \approx 2.15$

Step2: Calculate regression intercept

Use $\bar{y} = m\bar{x} + b$ to find $b$:
$b = \bar{y} - m\bar{x} = 41.97 - 2.15\times21.21 \approx 41.97 - 45.60 = -3.63$

Step3: Predict 2022 value for State13

Use regression equation $\hat{y} = mx + b$, substitute $x=23.8$:
$\hat{y} = 2.15\times23.8 - 3.63$
$\hat{y} = 51.17 - 3.63 = 47.54$

Answer:

$\approx 47.5$ (rounded to one decimal place, or 47.54 for precise value)